We numerically study the vortex-vortex interaction in multi-component homogeneous BoseEinstein condensates within the realm of the Gross-Pitaevskii theory. We provide strong evidences that pairwise vortex interaction captures the underlying mechanisms which determine the geometric configuration of the vortices, such as different lattices in many-vortex states, as well as the bound vortex states with two (dimer) or three (trimer) vortices. Specifically, we discuss and apply our theoretical approach to investigate intra-and inter-component vortex-vortex interactions in two-and three-component Bose-Einstein condensates, thereby shedding light on the formation of the exotic vortex configurations. These results correlate with current experimental efforts in multi-component Bose-Einstein condensates, and the understanding of the role of vortex interactions in multiband superconductors.